How do I make my object look smooth without using the subdivision surface modifier?
I think most of the effect you're seeing comes from switching the default Shade Flat (which uses a single Normal per face when shading, from what I can tell) to Shade Smooth (which interpolates -- that is, smoothly averages -- between the face normals to get useful normal information in between the basic face normals).

For example, here's an Icosphere with Shade Flat on by default:



And here's what happens with Shade Smooth turned on (in the sub-menu indicated in the above image):



Secondarily, you may also find it useful to use the Smooth Vertex (from the Vertex sub-menu in Edit Mode) transformation to improve the smoothness of the model. I believe Daniel Kreuter, the YouTuber who made the tutorial you're following (I recognize the mesh-work, having followed the same tutorial myself in the past), makes liberal use of Smooth Vertex; he explains this in one of the tutorial's earliest videos (and that he likes to hotkey it to the 'Q' key).